Abstract:

The probability distribution P from which the history of our universe is
sampled represents a theory of everything or TOE. We assume P is formally
describable. Since most (uncountably many) distributions are not, this
imposes a strong inductive bias. We show that P(x) is small for any
universe x lacking a short description, and study the spectrum of TOEs
spanned by two Ps: P1 reflects the most compact constructive descriptions,
P2 the fastest way of computing all computable objects. P1 requires
generalizations of traditional computability, Solomonoff's algorithmic
probability, and Kolmogorov complexity; it leads to describable objects
more random than Chaitin's "number of wisdom" Omega. P2 derives from
Levin's universal search in program space and a natural resource-oriented
postulate: the cumulative prior probability of all x incomputable within
time t by this optimal algorithm should be 1/t. Between P1 and P2 we find
a universal cumulatively enumerable measure that dominates traditional
enumerable measures; any such CEM must assign low probability to any
universe lacking a short enumerating program. We derive P-specific
consequences for observers evolving in computable universes, inductive
reasoning, quantum physics, philosophy, and the expected duration of
our universe.

Note:
This is a slightly revised version of a recent preprint
[#!Schmidhuber:00version1!#]. The essential results should be of
interest from a purely theoretical point of view independent of the
motivation through formally describable universes. To get to the
meat of the paper, skip the introduction and go immediately to
Subsection 1.1 which provides a condensed outline of the main
theorems.